Homework 4

Directions:

1. Show each step of your work.
2. Turn in your answers in class on a physical piece of paper.
3. Staple multiple sheets together.
4. Feel free to use Desmos for graphing.

Answer the following:

1. Find the derivative with respect to the correct variable for each of the following functions. Derivative rules are allowed.
   1. $f(x) = (-3)^2$
   2. $f(x) = x^{2.9}$
   3. $f(r) = -9r^{2/3}$
   4. $f(x) = 5x^2 + 6x + 7$
   5. $f(r) = \dfrac{r^5 + 2r^4 + 3r + 1}{r}$
   6. $f(a) = \dfrac{5}{a^4} - \dfrac{2}{\sqrt[3]{a^2}} -\dfrac{1}{a} + 200$
   7. $f(x) = \dfrac{5}{x^4} + \dfrac{\sqrt{x^3}}{3} + x$
   8. $f(x) = 2x\left(3x^2 + 1\right)$
   9. $f(t) = \left(t^2+t+1\right)\left(t-1\right)$
   10. $f(x) = \left(x^3 + 2x^2 + 3x + 4\right)\left(x - \dfrac{2}{x^2}\right)$
   11. $f(a) = \dfrac{1}{a^2 + 1}$
   12. $f(t) = \dfrac{t + 1}{t^2 + 1}$
   13. $f(x) = \dfrac{x + \sqrt[5]{x^4}}{2x + 3}$
   14. $f(x) = (2x-1)^7$
   15. $f(x) = \dfrac{1}{(4x+3)^3}$
   16. $f(x) = \sqrt[3]{(x^2 + 3)^4}$
   17. $f(t) = \sqrt{t + 1}\sqrt[3]{t + 2}$
2. You started a company and hired an analyst to model your total revenue in your first two years of operations. The analyst says the total revenue seems to follow \[f(t) = \dfrac{0.4t^3}{1 + 0.4t^2} \qquad 0 \leq t \leq 2\] where $f(t)$ is measured in millions of dollars and $t=0$ is the date your company was started.
   
   1. What is your company's total revenue in the beginning of the second year of operation?
   2. How fast were your company's sales increasing at the beginning of the second year of operation?
   3. The analyst's model was slightly incorrect and says the sales data is better modeled by \[g(t) = \dfrac{0.4t^3}{1.2 + 0.4t^2} \qquad 0 \leq t \leq 2\] Answer the previous two parts using this new model.
   4. Given this new model, how far off was the analyst initially?
3. Recall the last question in Lecture 2.2.
   The concentration of carbon monoxide in the air due to automobile exhaust $t$ years from now is estimated to be \[f(t) = 0.01(0.2t^2 + 4t + 64)^{2/3}\] Find the rate at which the level of carbon monoxide is changing with respect to time.